{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:BNTW4AXZQSYA4SW6G5H6FM6RAZ","short_pith_number":"pith:BNTW4AXZ","canonical_record":{"source":{"id":"1309.6539","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-09-25T15:03:30Z","cross_cats_sorted":[],"title_canon_sha256":"2879908e7cdd1f0b3368b186178964e3e30dcf0970de47f1bd798c0b40d111a8","abstract_canon_sha256":"acd801754b1d4f3389b1ee79c2f6d670b7f8c1f8b4d566892a92de5bbefc6f11"},"schema_version":"1.0"},"canonical_sha256":"0b676e02f984b00e4ade374fe2b3d10662f17df480fffe0711bb31300999f3c5","source":{"kind":"arxiv","id":"1309.6539","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.6539","created_at":"2026-05-18T02:20:00Z"},{"alias_kind":"arxiv_version","alias_value":"1309.6539v4","created_at":"2026-05-18T02:20:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.6539","created_at":"2026-05-18T02:20:00Z"},{"alias_kind":"pith_short_12","alias_value":"BNTW4AXZQSYA","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"BNTW4AXZQSYA4SW6","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"BNTW4AXZ","created_at":"2026-05-18T12:27:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:BNTW4AXZQSYA4SW6G5H6FM6RAZ","target":"record","payload":{"canonical_record":{"source":{"id":"1309.6539","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-09-25T15:03:30Z","cross_cats_sorted":[],"title_canon_sha256":"2879908e7cdd1f0b3368b186178964e3e30dcf0970de47f1bd798c0b40d111a8","abstract_canon_sha256":"acd801754b1d4f3389b1ee79c2f6d670b7f8c1f8b4d566892a92de5bbefc6f11"},"schema_version":"1.0"},"canonical_sha256":"0b676e02f984b00e4ade374fe2b3d10662f17df480fffe0711bb31300999f3c5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:00.419834Z","signature_b64":"5d/qe4ORGy1sk/72o1rQC7McrEOgOE6ywuwtcws8HatJYNp4pykWe5hcmfFJPmKsP86hRZMrm3lw092XmRxpBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b676e02f984b00e4ade374fe2b3d10662f17df480fffe0711bb31300999f3c5","last_reissued_at":"2026-05-18T02:20:00.419142Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:00.419142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.6539","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:20:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OWMGxyArdsWZPlA5sDP5FYqK/LgUpljM/O9fOZ0QjIY42pxEJFeR8OqLmCcQw/xX5ytxVmpvbsGGR+VJZ9SbBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T11:25:12.771685Z"},"content_sha256":"7fc5982975488f93061e5e7cfe0214b4221f64d35560a28c65aaef6e11251e95","schema_version":"1.0","event_id":"sha256:7fc5982975488f93061e5e7cfe0214b4221f64d35560a28c65aaef6e11251e95"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:BNTW4AXZQSYA4SW6G5H6FM6RAZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the ergodicity of geodesic flows on surfaces of nonpositive curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Weisheng Wu","submitted_at":"2013-09-25T15:03:30Z","abstract_excerpt":"Let $M$ be a smooth compact surface of nonpositive curvature, with genus $\\geq 2$. We prove the ergodicity of the geodesic flow on the unit tangent bundle of $M$ with respect to the Liouville measure under the condition that the set of points with negative curvature on $M$ has finitely many connected components. Under the same condition, we prove that a non closed \"flat\" geodesic doesn't exist, and moreover, there are at most finitely many flat strips, and at most finitely many isolated closed \"flat\" geodesics."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6539","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:20:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KEic5JyUb7JmCQ/EOqE0F3KKG0gaDm7SJmSbtFWSupg9eGe/OI083EDR7nFwl3LEpuY2aG2sAygH25KN+xNmDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T11:25:12.772014Z"},"content_sha256":"d11e8d5dae84d483eb6581b4fe00296f9daa508385171e060f4ce10491ee4cba","schema_version":"1.0","event_id":"sha256:d11e8d5dae84d483eb6581b4fe00296f9daa508385171e060f4ce10491ee4cba"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ/bundle.json","state_url":"https://pith.science/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-27T11:25:12Z","links":{"resolver":"https://pith.science/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ","bundle":"https://pith.science/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ/bundle.json","state":"https://pith.science/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BNTW4AXZQSYA4SW6G5H6FM6RAZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:BNTW4AXZQSYA4SW6G5H6FM6RAZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"acd801754b1d4f3389b1ee79c2f6d670b7f8c1f8b4d566892a92de5bbefc6f11","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-09-25T15:03:30Z","title_canon_sha256":"2879908e7cdd1f0b3368b186178964e3e30dcf0970de47f1bd798c0b40d111a8"},"schema_version":"1.0","source":{"id":"1309.6539","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.6539","created_at":"2026-05-18T02:20:00Z"},{"alias_kind":"arxiv_version","alias_value":"1309.6539v4","created_at":"2026-05-18T02:20:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.6539","created_at":"2026-05-18T02:20:00Z"},{"alias_kind":"pith_short_12","alias_value":"BNTW4AXZQSYA","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"BNTW4AXZQSYA4SW6","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"BNTW4AXZ","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:d11e8d5dae84d483eb6581b4fe00296f9daa508385171e060f4ce10491ee4cba","target":"graph","created_at":"2026-05-18T02:20:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $M$ be a smooth compact surface of nonpositive curvature, with genus $\\geq 2$. We prove the ergodicity of the geodesic flow on the unit tangent bundle of $M$ with respect to the Liouville measure under the condition that the set of points with negative curvature on $M$ has finitely many connected components. Under the same condition, we prove that a non closed \"flat\" geodesic doesn't exist, and moreover, there are at most finitely many flat strips, and at most finitely many isolated closed \"flat\" geodesics.","authors_text":"Weisheng Wu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-09-25T15:03:30Z","title":"On the ergodicity of geodesic flows on surfaces of nonpositive curvature"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.6539","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:7fc5982975488f93061e5e7cfe0214b4221f64d35560a28c65aaef6e11251e95","target":"record","created_at":"2026-05-18T02:20:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"acd801754b1d4f3389b1ee79c2f6d670b7f8c1f8b4d566892a92de5bbefc6f11","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-09-25T15:03:30Z","title_canon_sha256":"2879908e7cdd1f0b3368b186178964e3e30dcf0970de47f1bd798c0b40d111a8"},"schema_version":"1.0","source":{"id":"1309.6539","kind":"arxiv","version":4}},"canonical_sha256":"0b676e02f984b00e4ade374fe2b3d10662f17df480fffe0711bb31300999f3c5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0b676e02f984b00e4ade374fe2b3d10662f17df480fffe0711bb31300999f3c5","first_computed_at":"2026-05-18T02:20:00.419142Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:20:00.419142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5d/qe4ORGy1sk/72o1rQC7McrEOgOE6ywuwtcws8HatJYNp4pykWe5hcmfFJPmKsP86hRZMrm3lw092XmRxpBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:20:00.419834Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.6539","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7fc5982975488f93061e5e7cfe0214b4221f64d35560a28c65aaef6e11251e95","sha256:d11e8d5dae84d483eb6581b4fe00296f9daa508385171e060f4ce10491ee4cba"],"state_sha256":"c7ba17365a353d379556463885b757988b684b6020c468e6feff011a9bce2e14"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q0G/+xAFkn3rL7qaYTWJVt1/E6tgXB4pjUx0u63yoPtRQOduGh5dbZmlMzwd7fcDn7151sxUTDt3Vfinc/TPAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T11:25:12.773912Z","bundle_sha256":"0b0eed98b351d4f72732039d9ce263e8882c3adf56d228a6eb0d4adf8210e1de"}}